In the area of modern algebra known as group theory, a **Tarski monster group**, named for Alfred Tarski, is an infinite group *G*, such that every proper subgroup *H* of *G*, other than the identity subgroup, is a cyclic group of order a fixed prime number *p*. A Tarski monster group is necessarily simple. It was shown by Alexander Yu. Olshanskii in 1979 that Tarski groups exist, and that there is a Tarski *p*-group for every prime *p* > 10^{75}. They are a source of counterexamples to conjectures in group theory, most importantly to Burnside's problem and the von Neumann conjecture.

Let be a fixed prime number. An infinite group is called a Tarski Monster group for if every nontrivial subgroup (i.e. every subgroup other than 1 and G itself) has elements.

- is necessarily finitely generated. In fact it is generated by every two non-commuting elements.
- is simple. If and is any subgroup distinct from the subgroup would have elements.
- The construction of Olshanskii shows in fact that there are continuum-many non-isomorphic Tarski Monster groups for each prime .
- Tarski monster groups are an example of non-amenable groups not containing a free subgroup.